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Consider a monopolist setting a single price to all consumer that faces a demand curve of P = 30 – q, where P is price and q is the quantity sold. The monopolist has a marginal cost curve of MC = q. The government implements a per-unit tax of $9 per unit, to be paid for by the monopolist. What is the resulting deadweight loss?Group of answer choices

Question

Consider a monopolist setting a single price to all consumer that faces a demand curve of P = 30 – q, where P is price and q is the quantity sold. The monopolist has a marginal cost curve of MC = q. The government implements a per-unit tax of $9 per unit, to be paid for by the monopolist. What is the resulting deadweight loss?Group of answer choices

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Solution

To calculate the deadweight loss, we first need to find the quantity and price before and after the tax.

  1. Before the tax, the monopolist will set its marginal cost (MC) equal to marginal revenue (MR) to maximize profit. The demand curve is P = 30 - q, so the MR curve, which is the derivative of the total revenue (TR) curve, is MR = 30 - 2q. Setting MC = MR, we get q = 30 - 2q, which simplifies to 3q = 30, or q = 10. Substituting q = 10 into the demand curve, we get P = 30 - 10 = $20.

  2. After the tax, the monopolist's marginal cost increases by the amount of the tax, so the new MC curve is MC = q + 9. Setting this equal to the MR curve, we get q + 9 = 30 - 2q, which simplifies to 3q = 21, or q = 7. Substituting q = 7 into the demand curve, we get P = 30 - 7 = $23.

  3. The deadweight loss is the area of the triangle formed by the original and new quantity and the demand curve. The base of the triangle is the difference in quantity (10 - 7 = 3) and the height is the difference in price (2323 - 20 = 3).Sothedeadweightlossis0.5baseheight=0.533=3). So the deadweight loss is 0.5 * base * height = 0.5 * 3 * 3 = 4.5.

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